Answer
$k = \frac{3}{4}$
Work Step by Step
Step 1 To find the curvature at the given point $(3,0)$, we will use the relation between the osculating circle's radius $\rho$ and the curvature $k$ at a point given by: \[ k = \frac{1}{\rho} \] Step 2 We can compare the figure given with Figure 12.5.5, where we can see the radius $\rho$ of the osculating circle to the major axis is $\frac{4}{3}$. The vector equation of the ellipse in Figure 12.5.5 is $\mathbf{r}(t) = 2\cos(t)\mathbf{i} + 3\sin(t)\mathbf{j}$. The ellipse in this example has switched major and minor axes, and its equation is now $\mathbf{r}(t) = 3\cos(t)\mathbf{i} + 2\sin(t)\mathbf{j}$, but the osculating circle's radius is still $\rho = \frac{4}{3}$. Step 3 To find the curvature, we will plug in $\rho = \frac{4}{3}$: \[ k = \frac{1}{\rho} \Rightarrow k = \frac{1}{\frac{4}{3}} = \frac{3}{4} \] Result \[ k = \frac{3}{4} \]
